Converting Polar Equations to Rectangular Coordinates

Introduction to Polar and Rectangular Coordinates

Polar and rectangular (Cartesian) coordinate systems are two common ways to represent points and equations in the plane. Converting between these systems is a fundamental skill in trigonometry and precalculus, especially when analyzing curves and solving equations.

• Polar Coordinates: A point is represented as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis.

• Rectangular Coordinates: A point is represented as (x, y), where x and y are the horizontal and vertical distances from the origin, respectively.

Key Formulas for Conversion

• From Polar to Rectangular:

• From Rectangular to Polar:

Sample Problems and Solutions

The following examples demonstrate how to convert polar equations to rectangular form using the above formulas.

Example 1

• Given:

• Step 1: Recall that and .

• Step 2: Multiply both sides by if necessary to express in terms of and .

• Step 3: Substitute and .

• Result: becomes , which can be rearranged as .

Example 2

• Given:

• Step 1: Expand:

• Step 2: Multiply both sides by :

• Step 3: Substitute:

Example 3

• Given:

• Step 1: Multiply both sides by :

• Step 2: Substitute:

• Result: The rectangular equation is

Summary Table: Common Polar-Rectangular Conversions

Additional info: These conversions are essential for analyzing curves and solving equations in both coordinate systems, especially in trigonometry and calculus.